Time Series Models for Growth of Urban Population in SAARC Countries

Transcription

1 Advances in Managemen & Applied Economics, vol., no.1, 01, ISSN: (prin version), (online) Inernaional Scienific Press, 01 Time Series Models for Growh of Urban Populaion in SAARC Counries Md. Aikur Rahman Khan 1 and Md. Mosafizur Rahman Absrac The purpose of our presen sudy is o srike ou suiable models o explain he growh paern and o forecas for urban populaion in SAARC counries. Using he daa from UNPD for he years 1950 o 000 in five years inerval, we fied boh exponenial and models. We found he superioriy of models over exponenial models o explain he ime rend behavior of he urban populaion as a percenage of oal populaion and o forecas up o he year 05. We also found ha urbanizaion is faser in Bangladesh han any oher SAARC counries. JEL classificaion numbers: C10, R11 1 Deparmen of Populaion Science and Human Resource Developmen, Universiy of Rajshahi, Rajshahi-605, Bangladesh, khanmar77@yahoo.com Deparmen of Saisics, Saisics and Mahemaics School, Yunnan Universiy of Finance and Economics, Kunming-6501, P.R. China, mosafiz_bd1@yahoo.com Aricle Info: Received : December 5, 011. Revised : January 16, 01 Published online : February 8, 01

2 110 Urban Populaion in SAARC Counries Keywords: Urban Populaion, SAARC Counries, Roo Mean Squared Forecas Error (RMSFE), Cross Validiy Predicive Power (CVPP), Exponenial Growh and Auoregressive Moving Average () 1 Inroducion Time rend behavior of populaion parameer has been exensively sudied. To fi a ime rend model a firs graphical represenaion of he daa is examined. The line graph of populaion parameers generally posses he exponenial growh crierion (Misra, 1995; Bhende and Kanikar 1997). Thus, o explain he growh paern of populaion daa exponenial growh models were preferred by many auhors (UN, 1967 and 1997; UNFPA, 1993; BBS, 003). Bu, we can no assure ha he daa will be suiable for exponenial modeling raher here migh have some oher modeling echniques ha can be more suiable for. Time series daa are ime dependen and are auocorrelaed (Pankraz, 1991; Gujarai, 1995; Cleary and Hay, 1980). So, we can apply he ime series modeling echniques based on he effecs of auocorrelaion. In his paper, an aemp is made o examine he suiabiliy of boh exponenial and ime series models for he urban populaion in SAARC counries expressed as a percenage of oal populaion. Daa and Mehods The daa have been colleced from UNPD (1999) and Earhrend (004). The daa have been shown graphically using line plo (Figure1). We have analyzed he daa using some sophisicaed models including he ime rend behavior of he daa. As he ime frame was in five years inerval from 1950 o 000 so here migh have some impac over he passages of ime. In his recogniion we need o

3 Md. Aikur Rahman Khan and Md. Mosafizur Rahman 111 examine wheher he daa possess saionary crierion or no. We used he uni roo es of MacKinnon (1996) o es he saionariy of he daa. An auoregressive model of order p, AR(p) (Cleary and Hay, 1980; Pankraz, 1991, Gujarai, 1995; and Hamilon, 1994) is of he form Y α p i1 β Y u (1) i i where, Y is he value of he variable Y (urban populaion as a percenage of oal populaion) a ime, α is he inercep erm, β i (i 1,,..., p) are he parameers, Y is he i-h lagged variable, -i is he ime rend, γ is he coefficien of he ime variable, and u is he error erm which is a whie noise indeed. If we include he effec of q moving average erms, inercep, and rend componen along wih he AR(p) model hen an auoregressive moving average model of order p and q, ha is, (p,q/c,t) model (Gujarai, 1995 and Pankraz, 1991) can be formed as Y α p i1 β Y i i q j1 u j j u () where α is he inercep erm and is he ime rend. (p,q/c, T) refers o auoregressive moving average model wih p auoregressive erms and q moving average erms including inercep erm (C) and rend componen (T). For esing he significance of he fied model saionariy crierion of residuals and oulier deecion echniques (Pankraz, 1991) are applied. The saionariy of residuals are examined using he uni roo es (MacKinnon, 1996). We have used he mehod of examining sandardized residuals (absolue value of sandardized residuals over 3.0 implies he presence of an oulier) o examine he presence of oulier (Pankraz, 1991). We have also used he recen developed cross validiy predicive power (CVPP) and resriced cross validiy predicive power

4 11 Urban Populaion in SAARC Counries (RCVPP) (Khan and Ali, 003a). The cross validiy predicive power (CVPP) due o Sevens (1996) is (n 1)(n )(n 1) ρ cv 1- w(1- R ); w (3) n(n k 1)(n k ) and he resriced cross validiy predicive power is ρ 1- w(1 - R ); R = 0; oherwise. 1 w (n 1)(n )(n 1), w n(n k 1)(n k ), n k 1 (4) rcv where R is he coefficien of muliple deerminaion, n is he sample size, k is n 1 (n 1)(n -1)(n - ) he number of regressors used in he model, and w. n n Furher, he shrinkage of R in compuing RCVPP has been compued from he absolue difference beween RCVPP and 003a), ha is, η R (Sevens, 1996; and Khan and Ali, ρ rcv R 1 w(1 R ) R (5) Now, ρ rcv indicaes ha if we fi he same model o some oher daa from he same populaion hen he fied model will be able o explain 95% variaion of he dependen variable. Furher, η indicaes ha over he populaion he fied model is 99% sable. Thereafer, we have forecased from he fied models and have compued he roo mean squared forecas error o examine he forecasing performance of he fied model. Roo mean squared forecas error is compued (Pankraz, 1991) using he formula n r (O F ) n 1 RMSFE (6) r where O is he observed value, F is he corresponding forecased value, and r is he number of periods o forecas.

5 Md. Aikur Rahman Khan and Md. Mosafizur Rahman 113 Unforunaely, hese observed values, O s are no always in exisence. So, we have used he esimaed roo mean squared forecas error (ERMSFE) o examine he forecasing performance of he fied model (Khan and Ali, 003b) ha is compued as (nr)(nr k1)(nr k) SS(e) ERMSFE SS nr ()(1ρ rcv) r(n r 1)(n r )(n r 1) r (7) where, SS n r () is he sum of squared oal of n observed values and r forecased values, SS n () SS(oal) is he sum of squared oal of n observed values, SS(e) is he sum of squared residuals, and k is he number of predicors used in he model. 3 Resuls and Discussions In 1950 urban populaion of Bangladesh was 4.% of oal populaion. Bu, in 000 his percenage becomes 4.5, ha is, wihin 50 years he incremen of urban populaion is 0.3%. Wihin hese 50 years ( ) urban populaion in India, Pakisan, Sri Lanka, Nepal, Bhuan, and Maldives lifed up 11.1%, 19.5%, 9.%, 9.6%, 5.0%, and 15.5%, respecively. If we order he counry wih respec o he incremen of urban populaion wihin 1950 o 000 hen we can wrie BD>PK>ML>IN>NP>SL>BH. Thus, we can say ha he urbanizaion is faser in Bangladesh han any oher SAARC counries. To fi ime rend models we have examined he saionariy of he variables using he uni roo es and have found ha all he variables are saionary a level, ha is, inegraed of order zero. The esimaed resuls are given a Table 1.

6 114 Urban Populaion in SAARC Counries bd in pk nepal maldiv sril bhuan Figure1: Urban populaion (in percen of oal populaion) in SAARC counries Table 1: Uni Roo Tes Variables Specificaion DF-Value MacKinnon Criical Saionary a Bangladesh None * Level India None * Level Pakisan None * Level Nepal None * Level Bhuan None * Level Maldives C, T and lag ** Level Sri Lanka None ** Level Here * (**) indicaes values a 1% (5%) level. We have fied models o explain he ime rend behavior of hose variables (Table ). Similarly, he fied exponenial growh models have shown in Table 3. From Table and Table 3 we can say ha he compued shrinkage of he fied exponenial growh model for urban populaion in Bangladesh, Nepal, Bhuan, and Maldives are less han heir corresponding fied models.

7 Md. Aikur Rahman Khan and Md. Mosafizur Rahman 115 Similarly, he compued shrinkage of he fied model for urban populaion as a percenage of oal populaion in India, Pakisan, and Srilanka are less han heir corresponding fied exponenial growh models. Table : Fied models for SAARC counries Counry & Fied Models R RCVPP Shrinkage model Bangladesh T X (1,0/C) Prob. ( ) (0.0178) Invered AR 0.71 India T X (1,0/C,T) Pr ob. ( ) ( ) (0.0014) Invered AR 0.63 Pakisan T X (1,0/T) Prob. ( ) (0.0000) Invered AR 0.94 Sri Lanka T û (0,1/C,T) Pr ob. (0.0000) (0.000) (0.0000) Invered MA -.99 Nepal T û (0,1/T) Pr ob. ( ) (0.0000) Invered MA -.9 Bhuan T X (1,1/T) Pr ob. ( ) (0.0087) ( Invered AR 0.66, Invered MA 0. Maldives.49305T X (1,0/T) Prob. (0.0001) ( ) Invered AR Here RCVPP is resriced cross validiy predicive power. I is o noe ha he residual analysis (normaliy and oulier deecion) were performed. No oulier was presen in he daa se as all he sandardized residuals

8 116 Urban Populaion in SAARC Counries were wihin 3. We found ha he residuals were normal wih respec o heir normal probabiliy plo. For simpliciy of conen we would like o relax hose graphical represenaions. Table 3: Fied Exponenial growh models for SAARC counries Counry & Fied Models R RCVPP Shrinkage model Bangladesh T e Prob. = (0.0000) (0.0000) India T e Prob. = (0.0000) (0.0000) Pakisan T e sig. = (0.0000) (0.0000) Sri Lanka T e Prob. = (0.0000) (0.0011) Nepal T e Prob. = (0.0000) (0.0000) Bhuan T e Prob. = (0.0000) (0.0000) e Prob. = (0.0000) (0.0000) Maldives T Here RCVPP is resriced cross validiy predicive power Forecased urban populaion in SAARC counries have shown in Table 4 and in Table 5. Table 5 shows he forecased urban populaion as a percenage of oal populaion from he fied exponenial models (Table 3) and Table 4 shows he forecased values from he fied models (Table ). Also, he ERMSFE have been incorporaed in hose ables. From Table 4 and Table 5 we see ha he ERMSFEs from he fied models are comparaively smaller han hose from he fied exponenial growh models. Fied exponenial model for he urban

9 Md. Aikur Rahman Khan and Md. Mosafizur Rahman 117 populaion of Nepal and Bhuan have greaer RCVPP as well as greaer ERMSFE han he corresponding fied models. Bu, RMSFE decreases wih he increase of resriced cross validiy predicive power (as explained by Khan and Ali, 003b). Table 4: Forecased urban populaion using he fied models of Table Year Bangladesh India Pakisan Nepal Maldives Sri Lanka Bhuan (.5387) ( ) (1.4465) (0) (7.4813) (.58357) (0.4594) (.3171) (0.659) (1.493) (0.3368) (5.895) (.14494) ( ) (.39157) (0.636) (1.0350) (0.4105) ( ) ( ) ( ) (.47794) ( ) ( ) (0.4668) ( ) (1.9708) (0.3563) (.56901) (0.6163) (1.1999) ( ) (5.3740) ( ) ( ) Table 5: Forecased urban populaion using he fied models of Table 3 Year Bangladesh India Pakisan Nepal Maldives Sri Lanka Bhuan ( ) ( ) (1.391) ( ) (5.7090) (.96774) (0.4748) ( ) (1.0514) (1.588) ( ) (5.987) (.91585) ( ) (4.1841) ( ) ( ) ( ) ( ) ( ) (0.4960) ( ) ( ) ( ) ( ) (6.9148) ( ) ( ) ( ) (1.1163) ( ) ( ) ( ) ( ) ( ) Here, values in parenhesis are esimaed roo mean squared forecas error (ERMSFE).

10 118 Urban Populaion in SAARC Counries On he oher hand, he fied exponenial models for urban populaion of Bangladesh, Nepal, Bhuan, and Maldives posses less shrinkages bu more ERMSFEs compared o heir associaed fied models. Table 4 and Table 5 depic ha he ERMSFEs depend on boh he iniial populaion and he regression coefficien. If he iniial populaion is greaer bu he regression coefficien is smaller hen he ERMSFE becomes smaller and he ERMSFE becomes larger if he regression coefficien is larger. From Table 3 we find ha he regression coefficien of he fied exponenial model for Bangladesh is greaer han ha for Pakisan and India and so he ERMSFE for Bangladesh is larger. 5 Conclusions In fiing growh models for urban populaion in SAARC counries exponenial models for Bangladesh, Nepal, Bhuan, and Maldives lead less shrinkage bu more esimaed roo mean squared forecas error han he corresponding fied models. Bu, models for all he seven SAARC counries lead accepable shrinkage, smaller and more significanly acceped roo mean squared forecas error han he fied exponenial growh models. ACKNOWLEDGEMENTS. This research was suppored by Naional Naural Science Foundaion of China (YCT1017). References [1] Bangladesh Bureau of Saisics, Saisical Year Book, Various Issues. [] A.A. Bhende and T. Kanikar, Principles of Populaion Sudies, Himalay Publishing House, Mumbai, 1997.

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